On non-congruent numbers with $8a\pm 1$ type odd prime factors and tame kernels

Let $n$ be a positive square-free integer, where every odd prime factor of $n$ has form $8a\pm 1$. We determine when $n$ is non-congruent with second minimal $2$-primary Shafarevich-Tate group, in terms of the $4$-ranks of class groups and a Jacobi symbol. In particular, when every odd prime factor of $n$ has form $8a+1$, this condition is equivalent to the vanishing of the $4$-rank of the tame kernel of $\mathbb{Q}(\sqrt{n})$ for odd $n$, or $\mathbb Q(\sqrt{-n})$foreven$n$. This generalizes previous results.